Dressed jeff-1/2 objects in mixed-valence lacunar spinel molybdates

The lacunar-spinel chalcogenides exhibit magnetic centers in the form of transition-metal tetrahedra. On the basis of density-functional computations, the electronic ground state of an Mo413+ tetrahedron has been postulated as single-configuration a12 e4 t25, where a1, e, and t2 are symmetry-adapted linear combinations of single-site Mo t2g atomic orbitals. Here we unveil the many-body tetramer wave-function: we show that sizable correlations yield a weight of only 62% for the a12 e4 t25 configuration. While spin–orbit coupling within the peculiar valence orbital manifold is still effective, the expectation value of the spin–orbit operator and the g factors deviate from figures describing nominal t5 jeff = 1/2 moments. As such, our data documents the dressing of a spin–orbit jeff = 1/2 object with intra-tetramer excitations. Our results on the internal degrees of freedom of these magnetic moments provide a solid theoretical starting point in addressing the intriguing phase transitions observed at low temperatures in these materials.


SM-1. Embedded Cluster Approach
Both GaMo4S8 and GaMo4Se8 crystallize in a cubic fcc lattice with the space group F43m under high-temperature. 1 The main building unit is represented by a breathing pyrochlore lattice of the Mo sites, with alternating Mo4 tetrahedra connected via corners. The electronic structure of this particular Mo4 unit was studied using an embedded cluster model in a similar manner as described previously for GaNb4Se8/GaTa 4 Se8. 2 Starting from the respective experimentally determined structures for GaMo4S8 3 and GaMo4Se8 4 , a large array of more than 10 000 point charges was created using the EWALD program. 5,6 Here, initial charges of +3, +2.75, -1.5 and -2 for Ga, Mo, inner and outer X (with X = S, Se), respectively, were assumed.  Fig. 1 in the manuscript). Additionally, it was ensured that the electrostatic potential fitted charges of the quantum cluster closely resemble the initial charges given above. 7,8 The coordinates of point charge field and quantum cluster are given in the separate spreadsheets (zip file).

SM-2. Computational Details
The strongly correlated electronic structure within the Mo4 unit was investigated using the complete active space self-consistent field (CASSCF) method. 9 Missing dynamical correlation was accounted for by the N-electron valence 2 nd order perturbation theory (NEVPT2) method 10 with all internal orbitals correlated. Besides, the relativistic Douglas-Kroll-Hess (DKH) approximation 11 was enabled and the all-electron SARC-DKH-TZVPP 12 basis set for Mo and DKH-DEF2-TZVPP 13 basis set for S/Se were used to properly treat SOC. For further efficiency, the RIJCOSX approximation with automatic generation of Coulomb and Exchange auxiliary basis sets 14 for Mo and the DEF2/J 15 and DEF2-TZVPP/C 16 auxiliary basis sets for S/Se was applied. Those auxiliary basis functions were decontracted to be used together with the DKH method. Finally, the cECPs were modeled through pseudopotentials of Andrae et al. 17 for Mo and Bergner et al. 18 and Leininger et al. 19 for Ga and S. Calculations were performed with the ORCA program package, v5.0. 20

SM-3. Active Space Set-up
In accordance to our previous work, 2 Figure S1. The natural orbitals of the active space can be assigned according to Td point group symmetry to a leading configuration of a1 2 e 4 t2 5 t1 0 t2 0 (62%) in the ground state.
2 Figure S1. Twelve-orbital active space in molecular orbital representation. Notation according to Td point group symmetry.

Low-energy excitation energies of high-temperature (HT) phase
In the following Tables S1 and S2, the low-energy excitation energies of GaMo4S8 and GaMo4Se8 in the high-temperature (HT) cubic F43m phase on CASSCF(11e,12o) and NEVPT2 level of theory are listed. For both, also the influence of spin-orbit coupling (SOC) on the excited state energies are given. Focusing on GaMo4S8, the 2 T2 ground state is split by SOC into j = 1/2 and j = 3/2 effective states by 0.12 eV. The magnitude of this splitting is similar in CASSCF and NEVPT2, since the underlying wavefunction is not changed by NEVPT2. As a further effect of SOC, the following low-energy excitation energies are increased by about 0.08 eV through SOC. Table S1. Low-energy excitation energies (eV) of high-temperature (HT) [Mo4S16] 19-(CAS(11e,12o)). Four quartets and six doublets were included in the state-averaging procedure. Notation according to Td point group symmetry. In GaMo4Se8, the effect of SOC is of similar magnitude, since it mainly originates from the 4d orbitals of Mo. Generally, the low-energy excitation spectrum yields lower values by about 50 to 100 meV for the Se-coordinated lacunar spinel as compared to the Scoordinated one due to longer Mo-Se bond lengths leading to smaller crystal field splittings. Still, the ordering of excited states and the corresponding weight of the leading electronic configuration in the overall wavefunction of each excited state remain close to identical in both compounds, which presumably leads to similar magnetic properties.

Low-energy excitation energies of low-temperature (LT) phase
For the low-temperature (LT) phases of GaMo4S8 and GaMo4Se8 (Tables S3 and S4), triply degenerate T1/T2 states are split into A2/A1 and E terms as a result of the Jahn-Teller distortion. This effect is more pronounced as compared to the spin-orbit splitting in the HT phase (~300 vs. ~100 meV). Since SOC is not active for the 2 A1 ground state, the structure adopts an effective spin-1/2 under the influence of Jahn-Teller distortion. Interestingly, the underlying a1 2 e 4 e 4 a1 1 leading configuration of this state comes with a weight of only 62%, comparable to the 2 T2 ground state at high temperatures. Also, excited states show a decreasing weight of the leading configuration since more configurations mix into the overall wavefunction. By correcting the CASSCF values for dynamical correlation with the NEVPT2 approach a strong decrease in the excited state energies is observed for the S = 1/2 and most of S = 3/2 states. Still, there is some re-ordering of the excited state terms observed at ~2 eV, e.g. involving 2 A1 and 4 E states.

SM-5. Susceptibility Measurements on GaMo4S8
Temperature-dependent magnetic susceptibility curves across the structural transition were measured for single-crystalline GaMo4S8 using a SQUID magnetometer (Quantum Design MPMS3) along the cubic [100] direction with a magnetic field of 3 T (see Figure S 2 ). The phase transition is clearly visible as a kink in the inverse susceptibility at Ts = 45 K. A Curie-Weiss fit of the inverse susceptibility in the cubic high-temperature phase yields an effective paramagnetic moment of μeff = 2.05 μB and a Curie-Weiss temperature Θ = 0.22 K. This effective paramagnetic moment agrees well with the one found by Powell et al. 3 Besides, using the relation g=μ eff / √ S (S +1) and a spin of S = 1/2, a measured g factor of gexp = 2.37 is obtained, which exceeds the calculated one of gcalc = 2.18 for [Mo4S16] 19-(see manuscript). This deviation can be attributed to vibronic effects, that are not included in our calculations due to a rigid cluster model. At temperatures above 100 K, this is also reflected by the difference in the measured and calculated susceptibility data shown in Figure

SM-6. Different Low-Temperature (LT) GaMo4S8 Structures
In the manuscript, the crystal structure data of GaMo4S8 was taken from Powell et al. 3 Since deviations in the bond lengths/angles with respect to the measured crystal cannot be excluded, their effect on magnetic properties was studied. For this, we adopt the LT-GaMo4S8 crystal structure proposed by Francois et al., 25 which shows slightly longer Mo-Mo bond lengths of about 0.04 Å. The associated low-energy excitation energies are given in Table S5. Compared to the structure of Powell et al. 3 (see Table S3), the energies in Table S5 are considerably lower. This can be explained by the longer bonds of the latter, resulting in a decreased crystal field splitting in the Mo4 tetrahedron.
Associated with the decreased SOC eigenvalues in Table S5, the g factor becomes more anisotropic: for a CAS(11e,12o) we obtain g∥ = 1.58 and g⊥ = 2.67 using NEVPT2 (g∥ = 1.71 and g⊥ = 2.59 using CASSCF). As a consequence, the simulated magnetization curves (see Figure S4) also differ from the ones presented in Figure 2 of the manuscript. Most evident here is the lower [110] saturation magnetization value by about 0.07 μB, which still differs by 0.03 μB from the experimentally measured curve. From a theoretical perspective, it is therefore difficult to reach overall quantitative agreement, since slight deviations in the structure can be expected based on the experimental set-up.  19-in ORCA5.0 is given in the following Figure S4. In this particular example, the discussed high-temperature excited states (fourteen doublet and nine quartet roots) are requested in the CASSCF block as well as a subsequent treatment of SOC.